1. Field of the Invention
The invention relates to a non-recursive discrete filter for processing input signal samples x(m) in order to generate output signal samples y(n) occurring with a predetermined output sampling frequency f.sub.s, which discrete filter has a transfer characteristic with a central frequency f.sub.o. The filter transfer characteristic is related to the transfer characteristic of a predetermined low-pass filter having impulse response h.sub.l (i). The discrete filter comprises:
An input for receiving said input signal samples x(m); PA1 A first storage means for storing and supplying a given number of input signal samples x(m); PA1 A second storage means for storing and supplying filter coefficients a(i); PA1 A multiplying means for multiplying each input signal sample with a corresponding one of said filter coefficients for generating modified input signal samples z(n,i) = a(i) x(n-i) PA1 A first adding means; PA1 A means for coupling the input of the first adding means to the multiplying means. PA1 said output sampling frequency f.sub.s is related to said center frequency f.sub.o by the expression f.sub.s = 8f.sub.o ; PA1 said filter coefficients a(i) are given by the expressions EQU a(i) = h.sub.1 (i).multidot.1/2.sqroot.2 for i = odd (8) EQU a(i) = h.sub.1 (i) -- for i = even (9) PA1 said coupling means comprise first means for modifying the modified input signal samples z(n,i) with a coefficient sgn [cos(.pi. i/4)] for generating signal samples of the form z(n,i) sgn [cos (.pi.i/4)], said last signal samples being applied to the first adding means for generating first output signal samples y.sub.1 (n) of the form: ##EQU5## connected to the output of said multiplying means there are second means for modifying the input signal samples z(n,i) with a coefficient sgn [sin(.pi.i/4)] for generating signal samples of the form z(n,i).sgn [sin(.pi.i/4)]; PA1 second adder means to which the signal samples z(n,i). sgn [sin(.pi.i/4)] are applied for generating second output signal samples y.sub.2 (n) of the form: ##EQU6## The function sgn.multidot.[.alpha.] is defined as follows: ##EQU7## PA1 .vertline.cos(.pi. i/4).vertline. = .vertline.sin(.pi. i/4).vertline. for i = 1, 3, 5, 7, and that for the remaining values of i, namely i = 0, 2, 4, 6 that: PA1 .vertline.cos(.pi. i/4).vertline. = 1 and .vertline.sin(.pi. i/4).vertline. = 0, or .vertline.cos(.pi. i/4).vertline. = 0 and .vertline.sin(.pi. i/4).vertline. = 1.
2. Description of the Prior Art
A non-recursive discrete filter is an arrangement which is used for producing output signal samples y(n) which are each constituted by the sum of a number (N) of algebraic products which are related to N input signal samples x(n), y(n) and x(n) being related by the expression: ##EQU4## in which a(i) represent constant coefficients which are a function of the transfer characteristic of the desired filter.
Discrete filters can be classified in two groups (see chapter D, reference 1) namely in:
1. Sampled-data-filters. Such filters are adapted to process input signal samples x(n) and to produce output signal samples y(n) which can take on a continuum of values. In such a filter the first storage means for storing and supplying the input signal samples x(n) is, for example, formed by a shift register for non-amplitude discrete signal samples. This shift register may be constructed by means of, for example, "charge coupled devices" (CCD's).
2. Digital filters. The input signal samples x(n) and the output signal samples y(n) occurring in these filters as well as the filter coefficients are amplitude discrete and are available in the form of digital numbers with a given number of bits. The first storage means can again be constituted by a shift register wherein, however, each of the shift register elements is now arranged for storing and supplying a complete digital number x(n). A RAM (random access memory) may also be used as storage means. It should be noted that the way in which these digital numbers are represented (see also reference 1) is of no importance for the present invention.
Besides the two above-mentioned groups of discrete filters a third group also belongs to these discrete filters, namely the binary transversal filters. With these filters the input signal samples are amplitude-discrete but the filter coefficients and the output signal samples can take on a continuum of values (see reference 2).
From the preceding it follows that the expression (1) holds for all of the above-mentioned groups of discrete non-recursive filters. Consequently, the following applies to sampled-data filters, to digital filters as well as to binary transversal filters and the invention will be further explained with reference to a digital filter.
As mentioned above the filter coefficients a(i) are a function of the transfer characteristic of the desired filter. More in particular, for a low-pass filter having an impulse response h.sub.1 (i) these filter coefficients are given by the expression: EQU a(i) = h.sub.1 (i) (2)
As indicated in reference 3, in analog signal processing a bandpass filter having a given center, frequency f.sub.o and a bandwidth which is equal to 2f.sub.c can be derived from a lowpass filter having cut-off frequency f.sub.c and impulse response h.sub.1 (.tau.). If the impulse response of this bandpass filter is represented by h.sub.F (.tau.) then it holds that: EQU h.sub.F (.tau.) = h.sub.1 (.tau.) cos 2.pi.f.sub.o t (3)
In full agreement herewith, in discrete signal processing a bandpass discrete filter can be derived from a lowpass discrete filter having an impulse response h.sub.1 (i) and a bandwidth which is equal to half the bandwidth of the bandpass discrete filter. If h.sub.F (i) represents the impulse response of the bandpass discrete filter then it applies, for example, that: EQU h.sub.F (i) = h.sub.1 (i) cos (2.pi. if .sub.o /f.sub.s) (4)
The filter coefficients of this bandpass filter will be indicated by a.sub.F (i) and these coefficients are again given by the relation: EQU a.sub.F (i) = h.sub.F (i) (5)
The filter with the impulse response defined in expression (4) will be called an in-phase filter hereinafter.
In, for example, single sideband and vestigial sideband modulation systems (see, for example, references 4 and 5) not only is a filter having the impulse response defined in expression (4) used but also a filter having an impulse response of the form: EQU h.sub.Q (i) = h.sub.1 (i)sin (2.pi. if.sub.o /f.sub.s) (6)
The filter coefficients of this filter will be indicated by a.sub.Q (i) and are again given by the expression: EQU a.sub.Q (i) = h.sub.Q (i) (7)
The filter having the impulse response defined in expression (6) will be called quadrature filter hereinafter.
Using the discrete in-phase and quadrature filter in said modulation systems results, for example, in the use of two discrete filters which operate fully independently from one another. The in-phase filter makes use of the coefficients a.sub.F (i) and in the quadrature filter of the coefficients a.sub.Q (i). Another possibility is to use a filter which supplies within the sampling period T.sub.s = 1/f.sub.s two output signal samples in succession, wherein the first output signal sample is produced by using the coefficients a.sub.F (i) in expression (1) and wherein the second output signal sample is produced by using the coefficients a.sub.Q (i) in expression (1). In the first solution as well as in the second solution, simultaneously applying the in-phase and quadrature filter requires a storage capacity of such a size that all the coefficients a.sub.F (i) as well as all coefficients a.sub.Q (i) can be stored.